Sub-convexity problem for Rankin-Selberg $L$-functions

We establish a sub-convexity estimate for Rankin-Selberg $L$-functions in the combined level aspect, using the circle method. If $p$ and $q$ are distinct prime numbers, $f$ and $g$ are non-exceptional newforms (modular or Maass) for the congruence subgroups ${\mathrm{\Gamma }}_0(p)$ and ${\mathrm{\Gamma }}_0(q)$ (resp) with trivial nebentypus, then for all $ϵ>0$ we show that there exists an $A>0$ such that $$L(\frac{1}{2}+it,f\times g){\ll }_{ϵ,{\mu }_f,{\mu }_g}(1+\mathrm{\mid }t\mathrm{\mid }{)}^A\frac{(pq{)}^{1\mathrm{/}2+ϵ}}{\max \{p,q{\}}^{\frac{1}{64}}}\mathrm{.}$$ The dependence on ${\mu }_f$ and ${\mu }_g$, the parameters at infinity for $f$ and $g$ respectively, is polynomial. Further, if $p$ is fixed and $q \rightarrow \infty$,weimprovethisto$$ L\left(\frac{1}{2}+it, f \times g \right) \ll_{\epsilon,\mu_f,\mu_g}(p(1+|t|))^Aq^{\frac{1}{2}-\frac{1-2\theta}{27+28\theta}+\epsilon} , $$where$\theta$ is the exponent towards Ramanujan-conjecture for cuspidal automorphic forms. Unconditionally, we can take $\theta =7\mathrm{/}64$. This improves all previously known sub-convexity estimates in this case.